Refracted Ray Vector Formula

1. The Goal: From Angles to Vectors

Snell's Law gives us a relationship between angles, but our C++ code needs a direction vector $\vec{T}$ (for transmitted/refracted ray) given an incident vector $\vec{I}$ and a surface normal $\vec{N}$.

The core idea is to decompose the output vector $\vec{T}$ into components that we can calculate. A common technique in vector geometry, which you've likely used in graphics and physics, is to break a vector down into components parallel and perpendicular to a known vector. In our case, the surface normal $\vec{N}$ is the perfect reference.

Let's decompose $\vec{T}$ into two orthogonal vectors:

• A component parallel to the normal $\vec{N}$.
• A component perpendicular to the normal $\vec{N}$ (i.e., tangent to the surface).

As the diagram shows, we can write $\vec{T}$ as:
$\vec{T}=\vec{A}+\vec{B}$

Our task is to find expressions for $\vec{A}$ and $\vec{B}$ using our known quantities: $\vec{I}$, $\vec{N}$, ${\eta }_1$, and ${\eta }_2$. For this derivation, we will assume all vectors ($\vec{I}$, $\vec{N}$, $\vec{T}$) are of unit length.

2. Deriving the Vector Formula

Let's tackle each component, $\vec{A}$ and $\vec{B}$, one by one. The Scratchapixel article provides a fantastic, detailed walkthrough of this derivation. We'll summarize the key steps here.

Let's walk through the derivation's logic:

Step 1: Expressing the Components with Trigonometry

From the diagram and basic trigonometry:

• The component $\vec{B}$ is parallel to the normal $\vec{N}$ but points inward. Its length is $\cos ({\theta }_2)$. So, $\vec{B}=-\vec{N}\cos ({\theta }_2)$.
• The component $\vec{A}$ is perpendicular to the normal. Its length is $\sin ({\theta }_2)$. We need to find its direction.

Notice that the incident ray $\vec{I}$, the normal $\vec{N}$, and the refracted ray $\vec{T}$ all lie in the same plane (the "plane of incidence"). This means $\vec{A}$ must be in the same direction as the component of $\vec{I}$ that is perpendicular to $\vec{N}$.

We can find this direction vector, let's call it $\vec{M}$, by taking $\vec{I}$ and adding the part of it that is parallel to $\vec{N}$. The dot product $\vec{I}\cdot \vec{N}$ gives us $\cos ({\theta }_1)$. So, the vector parallel to $\vec{N}$ is $\vec{N}\cos ({\theta }_1)$.
Therefore, the tangential part is $\vec{I}+\vec{N}\cos ({\theta }_1)$. This vector has length $\sin ({\theta }_1)$. To get a unit vector $\vec{M}$, we divide by its length:
$\vec{M}=\frac{\vec{I}+\vec{N}\cos ({\theta }_1)}{\sin ({\theta }_1)}$
Now we can write $\vec{A}$:
$\vec{A}=\vec{M}\sin ({\theta }_2)=\frac{\vec{I}+\vec{N}\cos ({\theta }_1)}{\sin ({\theta }_1)}\sin ({\theta }_2)$

Step 2: Substituting and Simplifying with Snell's Law

Now we combine $\vec{A}$ and $\vec{B}$ and substitute what we know:
$\vec{T}=\vec{A}+\vec{B}=\left(\frac{\vec{I}+\vec{N}\cos ({\theta }_1)}{\sin ({\theta }_1)}\right)\sin ({\theta }_2)-\vec{N}\cos ({\theta }_2)$
This looks complicated, but we can simplify it using Snell's Law: $\frac{\sin ({\theta }_2)}{\sin ({\theta }_1)}=\frac{{\eta }_1}{{\eta }_2}$. Let's define the ratio $\eta =\frac{{\eta }_1}{{\eta }_2}$.
$\vec{T}=\eta (\vec{I}+\vec{N}\cos ({\theta }_1))-\vec{N}\cos ({\theta }_2)$
Rearranging terms gives a common form of the equation:
$\vec{T}=\eta \vec{I}+(\eta \cos ({\theta }_1)-\cos ({\theta }_2))\vec{N}$

Step 3: Eliminating Trigonometry

This is great, but it still contains $\cos ({\theta }_1)$ and $\cos ({\theta }_2)$. We want to replace them with dot products.

• $\cos ({\theta }_1)$ is simple. Since $\vec{I}$ and $\vec{N}$ are unit vectors, $\cos ({\theta }_1)=\vec{I}\cdot \vec{N}$. Note: To make sure the angle is acute, we often use $-\vec{I}\cdot \vec{N}$ if $\vec{I}$ points towards the surface. We'll handle this detail in the implementation. For now, let's just use $c_1=|\vec{I}\cdot \vec{N}|$.

• $\cos ({\theta }_2)$ is trickier. We use the identity ${\cos }^2(\theta )+{\sin }^2(\theta )=1$, so $\cos ({\theta }_2)=\sqrt{1-{\sin }^2({\theta }_2)}$.
  From Snell's law, we have $\sin ({\theta }_2)=\eta \sin ({\theta }_1)$.
  We also know ${\sin }^2({\theta }_1)=1-{\cos }^2({\theta }_1)=1-c_1^2$.
  Putting it all together:
  $\cos ({\theta }_2)=\sqrt{1-{\sin }^2({\theta }_2)}=\sqrt{1-{\eta }^2{\sin }^2({\theta }_1)}=\sqrt{1-{\eta }^2(1-c_1^2)}$

Substituting this back into our vector equation gives us the final formula, free of explicit angle calculations! The implementation in the PBR book and many other sources uses exactly this form.

3. The Condition for Total Internal Reflection

Our derived formula for $\cos ({\theta }_2)$ has an important side effect. Look at the term inside the square root:
$k=1-{\eta }^2(1-c_1^2)$
What happens if this term, $k$, is negative? We would be taking the square root of a negative number, which has no real solution.

This isn't a mathematical error; it's a signal of a physical phenomenon. This is precisely the condition for Total Internal Reflection (TIR) we discussed in the last lesson.

• This happens when ${\eta }^2(1-c_1^2)>1$.
• Since $\eta ={\eta }_1/{\eta }_2$, this condition is more likely when moving from a dense to a less-dense medium (${\eta }_1>{\eta }_2$, so $\eta >1$).
• Since $c_1=\cos ({\theta }_1)$, $1-c_1^2={\sin }^2({\theta }_1)$. The condition becomes ${\eta }^2{\sin }^2({\theta }_1)>1$, or $\sin ({\theta }_2)>1$, which is exactly what we identified as the trigger for TIR in the previous lesson.

When our code calculates this term and finds it's negative, it knows that no refraction occurs, and it must compute a reflection vector instead.

Conclusion

You've now seen how to get from the abstract physics of Snell's Law to a concrete, computable vector equation. This is a perfect example of the blend of physics, math, and engineering that makes ray tracing so compelling.

Key Takeaways:

• The refracted ray vector can be derived by decomposing it into components parallel and perpendicular to the surface normal.
• By combining this geometric construction with Snell's Law and trigonometric identities, we can create a formula that depends only on the incident vector, the normal, and the ratio of refractive indices.
• The final formula avoids expensive trig functions like acos or asin, using dot products instead, which is ideal for a high-performance C++ implementation.
• The math itself provides a natural test for Total Internal Reflection: if the term inside the square root becomes negative, refraction is physically impossible, and the ray must reflect.

Next Steps

We have the complete mathematical toolkit. In the next lesson, we will implement the logic to handle the phenomenon of total internal reflection. We'll take our derived formula and the TIR check and write a robust refract function in C++. This will also involve some important practical details, like correctly handling cases where a ray enters versus exits an object by swapping the refractive indices and flipping the normal.